It would be helpful to give a specific reference that uses this term in a way that you do not understand.
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Deane YangJun 29 '12 at 16:46

Sorry, I cannot find the exact paper. Just the rough context: For an equation $Au=f$, where $A$ is a differential operator(evolution type or not, Cauchy problem or not), $u,f$ belong to certain topological vector space(test function space $C_0^\infty$,Hilbert space, Banach space, or distribution space). A statement is put into this way: 'By duality argument, we can get the existence of u...' Sorry for the vagueness. Nevertheless I expect 'broad' answers from all aspects. I thank Mr.Anatoly's answer. But I'm not sure his answer is what I want.
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pde_bkJun 29 '12 at 18:52

2 Answers
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Of course the "duality argument" may have different meaning. However a frequent use of the term is as follows. For typical evolution equations, the correct solvability of the Cauchy problem in some space $\Phi$ implies the uniqueness of its solution in the dual space $\Phi'$. This approach gives precise uniqueness classes for equations and systems with constant coefficients. See

Such an argument would crucially use properties of the dual problem (e.g., a problem involving the adjoint operator). One can say that the weak formulation is a duality based approach, but typically you would expect a more substantial use of duality in a duality argument.